Partial Fractions Quotient And Remainder Of (x^2+8)/(x^2-5x+6) Explained

Hey there, math enthusiasts! Ever stumbled upon a rational expression that looks a bit intimidating? Don't worry, we've all been there. Today, we're going to break down a classic problem: finding the quotient, remainder, and partial fraction decomposition of the expression (x^2 + 8) / (x^2 - 5x + 6). This might sound like a mouthful, but trust me, it's a super useful skill to have in your mathematical toolkit. We'll go through each step together, making sure everything is crystal clear. So, let's dive in and conquer this algebraic challenge!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. The main goal here is to express the given rational function (x^2 + 8) / (x^2 - 5x + 6) in a different form. This form will involve a quotient (the result of the division), a remainder (what's left over after the division), and partial fractions (breaking down a complex fraction into simpler ones). Think of it like taking a mixed number, such as 7/5 and rewriting it as 1 + 2/5. We're doing something similar here, but with polynomials instead of numbers.

The expression we're working with is a rational function, which is simply a fraction where the numerator and denominator are polynomials. In this case, the numerator is x^2 + 8, and the denominator is x^2 - 5x + 6. Our mission is to perform polynomial long division and then decompose the resulting fraction into simpler partial fractions. This process is incredibly helpful in calculus, especially when dealing with integration problems. By breaking down complex rational functions into simpler terms, we can often find solutions more easily. So, this isn't just a theoretical exercise; it's a practical skill that will serve you well in more advanced math courses.

Partial fraction decomposition is a technique that allows us to rewrite a complex rational expression as a sum of simpler fractions. Each of these simpler fractions will have a denominator that is a factor of the original denominator. This is extremely useful in calculus, especially when integrating rational functions. The process involves several steps, including factoring the denominator, setting up the partial fraction decomposition, solving for the unknown coefficients, and finally, rewriting the original expression in terms of these simpler fractions. By mastering this technique, you'll be able to tackle a wide range of calculus problems with confidence.

Step-by-Step Solution

1. Polynomial Long Division

The first step in tackling this problem is to perform polynomial long division. This will help us find the quotient and remainder. Remember those long division problems you did back in elementary school? Well, this is the same concept, but with polynomials. We're dividing the numerator (x^2 + 8) by the denominator (x^2 - 5x + 6). The goal is to find out how many times the denominator "fits" into the numerator and what's left over.

Performing polynomial long division might seem daunting at first, but it's a systematic process that becomes easier with practice. First, set up the long division as you would with numbers, with the dividend (x^2 + 8) inside the division symbol and the divisor (x^2 - 5x + 6) outside. Then, divide the leading term of the dividend (x^2) by the leading term of the divisor (x^2). This gives you the first term of the quotient, which in this case is 1. Multiply the divisor by this term and subtract the result from the dividend. This will give you a new polynomial to work with. Repeat the process until the degree of the remainder is less than the degree of the divisor. This whole process allows us to separate the whole number part (the quotient) from the fractional part (the remainder over the divisor).

When we divide x^2 + 8 by x^2 - 5x + 6, we get a quotient of 1 and a remainder of 5x + 2. This means we can rewrite the original expression as 1 + (5x + 2) / (x^2 - 5x + 6). Notice how the quotient (1) is a whole number, and the remaining fraction has a numerator (5x + 2) and the original denominator (x^2 - 5x + 6). This is analogous to expressing an improper fraction (like 7/5) as a mixed number (1 + 2/5). The next step is to decompose the fractional part into partial fractions, which will give us a sum of simpler fractions.

2. Factoring the Denominator

Now, let's focus on the fractional part: (5x + 2) / (x^2 - 5x + 6). To decompose this into partial fractions, we first need to factor the denominator. Factoring the denominator is crucial because it tells us the form of the partial fractions we'll be looking for. The denominator is a quadratic expression, so we're looking for two factors that multiply to give the constant term (6) and add up to give the coefficient of the linear term (-5).

Factoring quadratic expressions is a fundamental skill in algebra. There are several techniques you can use, such as trial and error, using the quadratic formula, or completing the square. In this case, we're looking for two numbers that multiply to 6 and add to -5. A little bit of thought reveals that -2 and -3 fit the bill perfectly. Therefore, we can factor the denominator x^2 - 5x + 6 as (x - 2)(x - 3). This factorization is the key to setting up the partial fraction decomposition correctly.

By factoring the denominator as (x - 2)(x - 3), we've identified the two linear factors that will form the denominators of our partial fractions. This means we can express the fraction (5x + 2) / (x^2 - 5x + 6) as the sum of two fractions: A / (x - 2) and B / (x - 3), where A and B are constants that we need to determine. This decomposition allows us to break down the complex fraction into simpler pieces that are easier to work with.

3. Partial Fraction Decomposition Setup

With the denominator factored, we can set up the partial fraction decomposition. Since we have two linear factors, (x - 2) and (x - 3), we can write:

(5x + 2) / ((x - 2)(x - 3)) = A / (x - 2) + B / (x - 3)

where A and B are constants that we need to find. This equation is the foundation of our partial fraction decomposition. It states that the original fraction can be expressed as the sum of two simpler fractions, each with one of the linear factors as its denominator. The next step is to find the values of the constants A and B.

Setting up the partial fraction decomposition correctly is essential for solving the problem. The number and form of the partial fractions depend on the factors of the denominator. If the denominator has distinct linear factors (like in this case), then each factor will correspond to a partial fraction of the form A / (x - r), where r is the root of the factor. If there are repeated linear factors, or irreducible quadratic factors, the setup will be slightly different. Understanding these rules is crucial for handling various types of rational expressions.

Our goal now is to solve for the constants A and B. There are several methods we can use to do this, including the method of equating coefficients and the method of substituting values. Both methods rely on clearing the denominators in the equation and then solving the resulting algebraic equations. Once we've found the values of A and B, we can substitute them back into the partial fraction decomposition to get the final result.

4. Solving for the Constants

To find the values of A and B, we can multiply both sides of the equation by the common denominator, (x - 2)(x - 3):

5x + 2 = A(x - 3) + B(x - 2)

Now, we can use a couple of methods to solve for A and B. One method is to substitute specific values of x that will eliminate one of the variables. For example, if we let x = 2, the term with B will become zero, and we can solve for A. Similarly, if we let x = 3, the term with A will become zero, and we can solve for B.

Solving for the constants A and B is a crucial step in the partial fraction decomposition process. The method of substituting values is often the most efficient way to find these constants, especially when dealing with distinct linear factors. By strategically choosing values of x that make one of the factors zero, we can isolate the other variable and solve for it. This approach simplifies the algebra and allows us to find the constants quickly and accurately.

Let's apply this method. If we let x = 2, we get:

5(2) + 2 = A(2 - 3) + B(2 - 2)

12 = -A

A = -12

And if we let x = 3, we get:

5(3) + 2 = A(3 - 3) + B(3 - 2)

17 = B

So, we've found that A = -12 and B = 17.

5. Writing the Partial Fractions

Now that we have the values of A and B, we can write the partial fraction decomposition:

(5x + 2) / ((x - 2)(x - 3)) = -12 / (x - 2) + 17 / (x - 3)

This tells us that the fraction (5x + 2) / (x^2 - 5x + 6) can be broken down into two simpler fractions: -12 / (x - 2) and 17 / (x - 3). This decomposition is incredibly useful for various mathematical operations, such as integration and finding limits.

Writing the partial fractions is the culmination of the decomposition process. It's where we take the constants we've found and substitute them back into the setup we established earlier. This gives us the final form of the partial fraction decomposition, which expresses the original fraction as a sum of simpler fractions. This form is often much easier to work with, especially in calculus applications.

6. Combining with the Quotient

Finally, we combine the quotient we found in the first step with the partial fractions:

(x^2 + 8) / (x^2 - 5x + 6) = 1 + 17 / (x - 3) - 12 / (x - 2)

And that's it! We've successfully found the quotient, remainder, and partial fraction decomposition of the given expression. This final form clearly shows the quotient (1) and the partial fractions, making it easier to analyze and manipulate the expression.

Combining the quotient with the partial fractions is the final step in solving the problem. It's essential to remember the quotient we obtained from the polynomial long division, as it forms an integral part of the final answer. By adding the quotient to the partial fraction decomposition, we get the complete representation of the original rational expression in a different form. This form is often more convenient for further calculations and analysis.

The Answer

Comparing our result with the given options, we see that the correct answer is:

C. 1 + 17/(x - 3) - 12/(x - 2)

This option matches the quotient and partial fractions we calculated, confirming our step-by-step solution. Choosing the correct answer is the final validation of our work. It's always a good idea to double-check your calculations and make sure your answer aligns with the options provided.

Key Takeaways

This problem demonstrates a powerful technique for simplifying rational expressions. By performing polynomial long division and then decomposing the resulting fraction into partial fractions, we can rewrite complex expressions in a more manageable form. This skill is invaluable in calculus and other areas of mathematics. Remember, guys, practice makes perfect! The more you work through problems like this, the more comfortable you'll become with the process.

Mastering partial fraction decomposition opens up a world of possibilities in calculus and beyond. It allows you to tackle complex integrals, solve differential equations, and analyze rational functions more effectively. The process involves several key steps, including polynomial long division, factoring the denominator, setting up the partial fraction decomposition, solving for the constants, and writing the final result. By understanding each step and practicing regularly, you can develop a strong foundation in this essential technique.

So, next time you encounter a daunting rational expression, remember the steps we've covered today. Break it down, factor, decompose, and conquer! You've got this!